A note on compact-like semitopological groups

Abstract

The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group G which is not T 3 . We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular T 1 compact and T 2 sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group (G, τ ) is a topological group provided there exists a Hausdorff topology σ ⊃ τ on G such that (G, σ) is a precompact topological group and (G, τ ) is weakly semiregular or (G, σ) is a feebly compact paratopological group and (G, τ ) is T 3 .

Some three-space properties of semitopological and paratopological groups

“…(2) There exists a Hausdorff sequentially compact ∞-semitopological group G which is not a paratopological group, see [13,Example 3].…”

A corrigendum to "A note on compact-like semitopological groups" [Carpathian Math. Publ. 2019, 11 (2), 442-452]

“…Unfortunately, lemma's proof from [1] contains an error. Namely, the inclusion π(U) ⊂ ππ −1 (V) fails, for instance, when π is the identity map and…”

“…Fortunately, in the paper [1] Lemma 3 is applied only once, namely in conjunction with Lemma 1 to prove Proposition 2. This application can be fixed because the map π considered in Lemma 1 satisfies a condition π −1 (π(U)) = U for every regular open subset of X.…”